A list of mathematical notions
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basic notions
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set, collection, classmember of
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set intersection
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set union
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set equality (extensional)
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subset
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set of sets
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empty set
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complement of a set
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ordered pair
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binary mapping, injective, bijective, surjective
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isomorphism
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cartesian product of sets
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n-ary function
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relations on sets (subsets of the cartesian product)
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relation
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equivalence relation
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transitive relation
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R (a relation) is reflexive, symmetric, antisymmetric, connected,
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projection map
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P is a partially/totally ordered set
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inf(a,b ) for a, b in P
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sup(a,b) for a, b in P
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ordinal number
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infinity
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infinitesimal? epsilon?
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domain of a function
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range of a function
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image of a function
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graph of a function
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binary operation
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associativity of a binary operation
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congruence with respect to an operation
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logic
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connectives:
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first-order quantifiers:
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for all x there exists y such that y = x2
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cardinality statements:
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there are at least two elements....,
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there are infinitely many prime numbers
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true (the constant)
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false (the constant)
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dunno (3-valued logic)
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is there an element ... (question)
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fuzzy-truth
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these are questionable but we add them nevertheless:
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T is a consequence of H
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a formula F is valid
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a formula F is satisfiable
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formula F and formula G are logically equivalent
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m is a model of F
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substitution
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inference rule
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elimination rule
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introduction rule
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sequence
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axiom
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consistent set of formulas
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inconsistent set of formulas
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maximally consistent set of formulas
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countable set of formulas
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CNF form
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DNF form
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Horn clause
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existential closure
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universal closure
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arithmetic
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g = lcm(n,m)
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G = gcd(n,m)
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factorize(n), n integer
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integers mod n
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Euler phi function
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factorize( x2+ 5x +6)
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if "n divides m" .... (:= n !=0
/\ IntegerRemainder(n,m)=0)
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let p be prime
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solve(x2-1 / (x-1)> 0 and x +7 <=0)
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solve(1/2Log(x +8) + 1/3Log(x) = Log 6,x)
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simplify(Radical(a^m^2 * b^ mn / c ^ (mn^2), mn) ) (?)
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linear algebra
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matrix
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vector
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diagonalize(A)
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diagonal of A
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A is a square matrix
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U is the identity matrix
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minors(A)
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determinant(A)
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trace(A)
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lambda-matrix(A)
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LU decomposition(A)
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Hermitian_conjugate(A)
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transpose(A)
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the characteristic polynomial of a matrix (the coefficients of the polynomial
and the matrix come from the same ring)
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the companion matrix of a polynomial
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M is a triangular matrix with entries from the ring R
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Cayley Hamilton: A satisfies its characteristic equation
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calculus
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sin(x2+1)
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integrate(exp x,x)
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RootOf(X3+X+1 = 0)
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f is continuous
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lim(f(x), x=p)
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f(n) = (-1)n
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differentiate(differentiate(f(x,y),x),y)
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Sum(f(xi), i=1..n)
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continuous fractions
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root_multiplicity
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Hessian determinant(f)
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combinatorics
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permutation(n)
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combination(n,r)
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binomial(n,k)
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factorial(n)
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a graph
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a block design
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statistics
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algebraic constructions
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Let k be a field and V a vector space over k
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Let V be a vector space of dimension n over the Reals
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S is an algebraic structure
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S is a substructure
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P is a poset
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L is a lattice
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L is a distributive lattice
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S is a semigroup
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M is a monoid
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u is the identity w.r.t. operation f
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product of monoids M1 and M2
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G is a group
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G is an abelian group
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subgroup
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normal subgroup
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permutation group
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group presented by generators and relations
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matrix group
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homotopy group
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H is a subgroup of G
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H is a normal subgroup of G
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R is a ring
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R is a ring with unit
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R is a commutative ring
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S is a skew field
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R is an integral ring
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D is an integral domain (it is integral and commutative)
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R is a commutative ring with unit
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D is an integral unit domain (unit_ring /\ integral_domain)
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E is a Euclidean Domain
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F is a field
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let M be the ring of matrices over Q
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field K of prime characteristic p
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field of quotients of an integral domain D
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finitely generated R-module
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left R-module
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R-bimodule
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vector space over F
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algebra over a field F
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associative algebra over a field F
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exterior algebra over a vector space V over a field F
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commutative algebra over a field F
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Lie algebra over a field F
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Jordan algebra over a field F
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Miscellaneous
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category
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functor
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topology
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variety
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Lie group
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power series
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rational function
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Jacobi identity
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enveloping algebra
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universal object
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commutative diagram
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subspace
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algorithm
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identity (an equality that holds for all elements in a structure)
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projective space
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interval